arXiv subjects

Ahmad Biniaz

Publications and source records attributed to Ahmad Biniaz.

44 records · Page 3Linked to original sources

arXiv2015

Plane Bichromatic Trees of Low Degree

Let $R$ and $B$ be two disjoint sets of points in the plane such that $|B|\leqslant |R|$, and no three points of $R\cup B$ are collinear. We show that the geometric complete bipartite graph $K(R,B)$ contains a non-crossing spanning tree whose maximum degree is at most $\max\left\{3, \left\lceil \frac{|R|-1}{|B|}\right\rceil + 1\right\}$; this is the best possible upper bound on the maximum degree. This solves an open problem posed by Abellanas et al. at the Graph Drawing Symposium, 1996.

cs.CG↗

arXiv2015

Approximating the Bottleneck Plane Perfect Matching of a Point Set

A bottleneck plane perfect matching of a set of $n$ points in $\mathbb{R}^2$ is defined to be a perfect non-crossing matching that minimizes the length of the longest edge; the length of this longest edge is known as {\em bottleneck}. The problem of computing a bottleneck plane perfect matching has been proved to be NP-hard. We present an algorithm that computes a bottleneck plane matching of size at least $\frac{n}{5}$ in $O(n \log^2 n)$-time. Then we extend our idea toward an $O(n\log n)$-time approximation algorithm which computes a plane matching of size at least $\frac{2n}{5}$ whose edges have length at most $\sqrt{2}+\sqrt{3}$ times the bottleneck.

cs.CG↗

arXiv2015

Strong Matching of Points with Geometric Shapes

Let $P$ be a set of $n$ points in general position in the plane. Given a convex geometric shape $S$, a geometric graph $G_S(P)$ on $P$ is defined to have an edge between two points if and only if there exists an empty homothet of $S$ having the two points on its boundary. A matching in $G_S(P)$ is said to be $strong$, if the homothests of $S$ representing the edges of the matching, are pairwise disjoint, i.e., do not share any point in the plane. We consider the problem of computing a strong matching in $G_S(P)$, where $S$ is a diametral-disk, an equilateral-triangle, or a square. We present an algorithm which computes a strong matching in $G_S(P)$; if $S$ is a diametral-disk, then it computes a strong matching of size at least $\lceil \frac{n-1}{17} \rceil$, and if $S$ is an equilateral-triangle, then it computes a strong matching of size at least $\lceil \frac{n-1}{9} \rceil$. If $S$ can be a downward or an upward equilateral-triangle, we compute a strong matching of size at least $\lceil \frac{n-1}{4} \rceil$ in $G_S(P)$. When $S$ is an axis-aligned square we compute a strong matching of size $\lceil \frac{n-1}{4} \rceil$ in $G_S(P)$, which improves the previous lower bound of $\lceil \frac{n}{5} \rceil$.

cs.CG↗

arXiv2015

Packing Plane Perfect Matchings into a Point Set

Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? We prove that at least $\lceil\log_2{n}\rceil-2$ plane perfect matchings can be packed into any point set $P$. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem.

cs.CG↗

arXiv2014

Matching in Gabriel Graphs

Given a set $P$ of $n$ points in the plane, the order-$k$ Gabriel graph on $P$, denoted by $k$-$GG$, has an edge between two points $p$ and $q$ if and only if the closed disk with diameter $pq$ contains at most $k$ points of $P$, excluding $p$ and $q$. We study matching problems in $k$-$GG$ graphs. We show that a Euclidean bottleneck perfect matching of $P$ is contained in $10$-$GG$, but $8$-$GG$ may not have any Euclidean bottleneck perfect matching. In addition we show that $0$-$GG$ has a matching of size at least $\frac{n-1}{4}$ and this bound is tight. We also prove that $1$-$GG$ has a matching of size at least $\frac{2(n-1)}{5}$ and $2$-$GG$ has a perfect matching. Finally we consider the problem of blocking the edges of $k$-$GG$.

cs.CG↗

arXiv2014

Higher-Order Triangular-Distance Delaunay Graphs: Graph-Theoretical Properties

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set $P$ of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle $\triangledown$, and there is an edge between two points in $P$ if and only if there is an empty homothet of $\triangledown$ having the two points on its boundary. We consider higher-order triangular-distance Delaunay graphs, namely $k$-TD, which contains an edge between two points if the interior of the homothet of $\triangledown$ having the two points on its boundary contains at most $k$ points of $P$. We consider the connectivity, Hamiltonicity and perfect-matching admissibility of $k$-TD. Finally we consider the problem of blocking the edges of $k$-TD.

cs.CG↗

arXiv2013

An Optimal Algorithm for the Euclidean Bottleneck Full Steiner Tree Problem

Let $P$ and $S$ be two disjoint sets of $n$ and $m$ points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to $S$, in which each point of $P$ is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in $O((n+m)\log m)$ time, improving the previously best result by a logarithmic factor. We also prove a matching lower bound in the algebraic computation tree model.

cs.CG↗

arXiv2012

Fixed-Orientation Equilateral Triangle Matching of Point Sets

Given a point set $P$ and a class $\mathcal{C}$ of geometric objects, $G_\mathcal{C}(P)$ is a geometric graph with vertex set $P$ such that any two vertices $p$ and $q$ are adjacent if and only if there is some $C \in \mathcal{C}$ containing both $p$ and $q$ but no other points from $P$. We study $G_{\bigtriangledown}(P)$ graphs where $\bigtriangledown$ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-$Θ_6$ graphs and TD-Delaunay graphs. The main result in our paper is that for point sets $P$ in general position, $G_{\bigtriangledown}(P)$ always contains a matching of size at least $\lceil\frac{n-2}{3}\rceil$ and this bound cannot be improved above $\lceil\frac{n-1}{3}\rceil$. We also give some structural properties of $G_{\davidsstar}(P)$ graphs, where $\davidsstar$ is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of $G_{\davidsstar}(P)$ is simply a path. Through the equivalence of $G_{\davidsstar}(P)$ graphs with $Θ_6$ graphs, we also derive that any $Θ_6$ graph can have at most $5n-11$ edges, for point sets in general position.

cs.CG↗